Page 7 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series

Example 2

+
If a sequence with u n + 1 = 2u , n  Z , and u = 1 is given, find the first five terms of this sequence. State
n
1
the value of the n term in terms of n.
th
Solution: u n + 1 = 2u , n  Z .
+
n
u = 1
1
— 1
Hence, u = 2u = 2 2
1
2
1
1
— + —
u = 2u = 2 2 4
2
3
1
1
— + — + —
2 u = 2u = 2 1 8
2
4
3
4
1
1
1
1
— + — + — + —
4
2
8
u = 2u = 2 16
4
5
Hence, the first five terms of the sequence are
1
1
1
1
1
1
— 1 — + — 1 — + — + — 1 — + — + — + — 1
8
4
2
4
2
8
2
1, 2 , 2 2 4 , 2 and 2 16
— 1 — 3 — 7 15
—
4
8
16
2
or 1, 2 , 2 , 2 and 2 .
1
1
1
1
— + — + — + … + ––––––
2
4
8
th
The n term, u = 2 2 n – 1
n
A sequence with terms which are repeated after a certain fixed number of terms is known as a periodic sequence.
In trigonometry, the range of values for the graphs of sine and cosine functions are between –1 and 1. This
range repeats itself after an interval of 2π radians. For example,
+
the sequence with u = cos nπ , n  Z , has the terms 0, –1, 0, 1, … which repeats itself after four terms.
2
n
Therefore, this sequence is a periodic sequence.
n
Consider the sequence for u = (–1) , n  Z , which has the terms –1, 1, –1, 1, … This sequence oscillates
+
n
finitely between –1 and 1. Therefore, this sequence is also a periodic sequence.
n
The sequence for u = (–10) , n  Z , has terms –10, 100, –1000, 10 000, …, and oscillates infinitely between
+
n
–∞ and ∞. Therefore, this sequence is not a periodic sequence.

Convergent and divergent sequences
Consider the sequence
n
1.1, 1.01, 1.001, 1.0001, …, 1 + (0.1) , …
The n term of this sequence is
th
1
u = 1 + 1 2 n
n
10
1
1
When n → ∞, 1 2 n → 0 and 1 + 1 2 n → 1.
10
10
We write
1
n
lim u = lim 1 + 1 2 4 = 1.
3
x → ∞ n n → ∞ 10
We say that this sequence converges to the value 1.
lim
u exists, then this sequence is called a convergent
In general, if u is the n term of a sequence and n → ∞ n
th
n
sequence and the value of the limit of u is known as the limiting value or limit of sequence.
n
A sequence which is not convergent is called a divergent sequence.
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02 STPM Math T T1.indd 94 3/28/18 4:21 PM